A note on primitive $1-$normal elements over finite fields

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by  Lucas Reis

A note on primitive 1−normal elements over finite fields
LucasReis
7 DepartamentodeMatema´tica,UniversidadeFederaldeMinasGerais,UFMG,BeloHorizonte,
1 MG,30123-970,Brazil
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n
a
J

9 Abstract
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Let q be a prime power of a prime p, n a positive integer and Fqn the finite field
] with qn elements. The k−normal elements over finite fields were introduced and
T
characterizedbyHuczynskaetal(2013). Undertheconditionthatnisnotdivisible
N
by p,theyobtainedanexistenceresultonprimitive1−normalelementsofF over
. qn
h F forq> 2. Inthisnote,weextendtheirresulttotheexcluded caseq = 2.
t q
a
m Keywords: FiniteFields,NormalBasis,k-normalelements,Primitiveelements
[ 2010MSC:12E20,11T06

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v
1. Introduction
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LetF bethefinitefieldwithqn elements,whereqisaprimepowerandnisa
6 qn
5 positiveinteger. Recallthatanelementα ∈ F issaidtobenormaloverF ifA =
qn q
0 {α,αq,··· ,αqn−1}isabasisofF overF ;Aiscalledanormalbasis. Normalbasis
. qn q
1
are frequently used in cryptography and computer algebra systems; sometimes it
0
7 is useful to take normal basis composed by primitive elements, i.e., generators of
1 the multiplicative group F∗ . The Primitive Normal Basis Theorem states that for
qn
v: any extension field Fqn of Fq, there exists a basis composed by primitive normal
i elements;thisresultwasfirstprovedbyLenstraandSchoof[6]andaproofwithout
X
theuseofacomputerwaslattergivenin[3].
r
a Acharacterization ofnormalelements isgivenin([7],Theorem2.39): anele-
mentα ∈ F isnormaloverF ifandonlyifthepolynomials
qn q
n−1
g (x) := αqixn−1−i and xn−1,
α
Xi=0
arerelativelyprimeoverF . Motivatedbythischaracterization, in[5],theauthors
qn
introduce k−normalelements:
Emailaddress: [email protected](LucasReis)
PreprintsubmittedtoElsevier January23,2017
Definition1.1. Letα ∈ F∗ andg (x) = n−1αqi·xn−1−i. Wesaythatαisk−normal
qn α i=0
overFq ifthegreatest commondivisor oPf xn−1andgα(x)overFqn hasdegreek.
From definition, 0−normal elements correspond to normal elements in the
usual sense. In the same paper, the authors give a characterization of k−normal
elementsandfindaformulafortheirnumber. Also,theyobtainanexistenceresult
onprimtive1−normalelements:
Theorem 1.2 ([5], Theorem 5.10). Let q = pe be a prime power and n a positive
integer notdivisible by p. Assumethatn ≥ 6ifq ≥ 11andthatn ≥ 3if3 ≤ q ≤ 9.
Thenthereexistsaprimitive1−normalelementofF overF .
qn q
Theauthors proposeanextensionoftheabovetheorem forallpairs(q,n)with
n ≥ 2asaproblem ([5],Problem6.2);theyconjectured thatsuchelementsalways
exist. However, it was proved in [1] that for odd q > 3 and n = 2, there are no
primitive 1−normal elements of F over F . The aim of this note is to extend
q2 q
Theorem 1.2 to the case when q = 2 and n is odd. Essentially, we show that the
toolsusedin[5]toproveTheorem1.2canbeadaptedtothatcase.
2. Existenceofprimitive1−normalelementsoverF
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First, we present some definitions and results that will be useful in the rest of
thispaper.
Definition2.1.
(a) Let f(x)beamonicpolynomialwithcoefficientsinF . TheEulerPhiFunction
q
forpolynomials overF isgivenby
q
F [x] ∗
Φ (f) = q ,
q (cid:12) hfi ! (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
wherehfiistheidealgenerated by f((cid:12)x)inF [x(cid:12)].
(cid:12) q (cid:12)
(b) If t is a positive integer (or a monic polynomial over F ), W(t) denotes the
q
numberofsquare-free (monic)divisors oft.
We have an interesting formula for the number of k−normal elements over
finitefields:
Lemma2.2([5],Theorem3.5). Thenumber N ofk−normalelementsofF over
k qn
F isgivenby
q
Φ (h), (1)
q
hX|xn−1
deg(h)=n−k
wherethedivisorsaremonicandpolynomial divisionisoverF .
q
2
In particular, if n ≥ 2, the number of 1−normal elements of F over F is at
qn q
leastequaltoΦ (T),whereT = xn−1.
q x−1
2.1. Asieveinequality
The proof of Theorem 1.2 is based in an application of the Lenstra-Schoof
method, introduced in [6]; this method has been used frequently in the character-
ization of elements in finite fields with particular properties like being primitive,
normal and of zero-trace. For more details, see [3] and [5]. In particular, from
Corollary5.8of[5],wecaneasilydeducethefollowing:
Lemma2.3. Supposethatqisapowerofaprime p,n ≥ 2isapositiveintegernot
divisible by pandT(x) = xn−1. If
x−1
W(T)·W(qn−1) < qn/2−1, (2)
thenthereexist1−normalelementsofF overF .
qn q
Inequality (2) is an essential step in the proof of Theorem 1.2 and it was first
studied in[2]; under thecondition that n ≥ 6for q ≥ 11and n ≥ 3 for3 ≤ q ≤ 9,
this inequality is not true only for afinite number ofpairs (q,n) (see Theorem 4.5
of[2]). Hereweextendthestudy ofinequality (2)tothecasewhenq = 2andnis
odd. First,wehavethefollowing:
Proposition 2.4. Suppose that n ≥ 3 is odd and T(x) := xn−1 ∈ F [x]. Then
x−1 2
W(T) ≤ 2n+59.
Proof. For each 2 ≤ i ≤ 4, let s be the number of irreducible factors of degree i
i
dividingT(x). Sincenisodd,T(x)hasnolinearfactor. Byadirectverificationwe
seethatthenumberofirreducible polynomials overF ofdegrees2,3and4is1,2
2
and 3, respectively. Hence s ≤ 1,s ≤ 2 and s ≤ 3. In particular, the number of
2 3 4
irreducible factorsofT(x)overF isatmost
2
n−1−2s −3s −4s n−1+3s +2s + s
2 3 4 + s + s + s = 2 3 4.
2 3 4
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Since n−1+3s2+2s3+s4 ≤ n−1+3+4+3 = n+9,weconclude theproof.
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AccordingtoLemma7.5in[4],W(2n−1) < 27n+2 ifnisodd. Inparticular, we
obtainthefollowing:
Corollary 2.5. Suppose that n , 15 is odd, q = 2 and T(x) = xn−1 ∈ F [x]. For
x−1 2
n > 9,inequality (2)holds.
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Proof. Noticethat n+9+n+2< n−1forn≥ 31. FromProposition2.4andLemma
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7.5of[4],itfollowsthatinequality (2)holdsforoddn ≥ 31. Theremaining cases
canbeverifieddirectly.
Wearereadytostateandproveourresult:
Theorem 2.6. Suppose that n ≥ 3 is odd. Then there exist a primitive 1−normal
elementofF overF .
2n 2
Proof. AccordingtoLemma2.3andCorollary2.5,thisstatementistrueforn > 9
if n , 15. For the remaining cases n = 3,5,7,9 and 15 we use the following
argument. LetPbethenumberofprimitiveelementsofF and N thenumberof
2n 1
1−normalelementsofF overF ;ifP+N > 2n,thereexistsaprimitive1−normal
2n 2 1
element of F over F . Notice that P = ϕ(2n −1) and, according to Lemma 2.2,
2n 2
N ≥ Φ xn−1 . Byadirectcalculation weseethat
1 2 x−1
(cid:16) (cid:17)
xn−1
ϕ(2n−1)+Φ > 2n,
2
x−1 !
forn = 3,5,7,9and15. Thiscompletes theproof.
References
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mialsoverfinitefields,Journal ofAlgebraandItsApplications 16(2017).
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(2013)170-183.
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